Most analytical experiments produce measurement data which require to be presented, analysed, and interpreted in respect of the chemical phenomena being studied. For such data and related analysis to have validity, methods which can produce the interpretational information sought need to be utilised. Statistics provides such methods through the rich diversity of presentational and interpretational procedures avail able to aid scientists in their data collection and analysis so that information within the data can be turned into useful and meaningful scientific knowledge. Pioneering work on statistical concepts and principles began in the eighteenth century through Bayes, Bernoulli, Gauss, and Laplace. Individuals such as Francis Galton, Karl Pearson, Ronald Fisher, Egon Pearson, and Jerzy Neyman continued the development in the first half of the twentieth century. Development of many fundamental exploratory and inferential data analysis techniques stemmed from real biological problems such as Darwin’s theory of evolution, Mendel’s theory of genetic inheritance, and Fisher’s work on agri cultural experiments. In such problems, understanding and quantifica tion of the biological effects of intra and interspecies variation was vital to interpretation of the findings of the research. Statistical
Statistical Analysis Methods for Chemists A Software-based Approach Statistical Analysis Methods for Chemists A Software-based Approach William P Gardiner Department of Mathematics, Glasgow Caledonian University, Glasgow, UK Information Services ISBN 0-85404-549-X 0The Royal Society of Chemistry 1997 All rights reserved Apart from any fair dealing for the purposes of research or private study, or criticism or review as permitted under the terms of the UK Copyright, Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, without the prior permission in writing of The Royal Society of chemistry, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK,or in accordance with the terms of the licences issued by the appropriate Reproduction Rights Organization outside the UK.Enquiries concerning reproduction outside the terms stated here should be sent to The Royal Society of Chemistry at the address printed on this page Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 4WF, UK Typeset by Computape (Pickering) Ltd, Pickering, North Yorkshire, UK Printed and bound by Athenaeum Press Ltd, Gateshead, Tyne and Wear, UK Preface Chemists carry out experiments to help understand chemical phenomena, to monitor and develop new analytical procedures, and to investigate how different chemical factors such as temperature, concentration of catalyst, pH, storage conditions of experimental material, and analytical procedure used affect a chemical outcome All such forms of chemical experimentation generate data which require to be analysed and interpreted in respect of the goals of the experiment and with respect to the chemical factors which may be influencing the measured chemical outcome To translate chemical data into meaningful chemical knowledge, a chemist must be able to employ presentational and analysis tools to enable the data collected to be assessed for the chemical information they contain Statistical data analysis techniques provide such tools and as such should be an integral part of the design and analysis of applied chemical experiments irrespective of complexity of experiment A chemist should therefore be familiar with statistical techniques of both exploratory and inferential type if they are to design experiments to obtain the most relevant chemical information for their specified objectives and if they are to use the data collected to best advantage in the advancement of their knowledge of the chemical phenomena under investigation The purpose of this book is to develop chemists’ appreciation and understanding of statistical usage and to equip them with the ability to apply statistical methods and reasoning as an integral aspect of analysis and interpretation of chemical data generated from experiments The theme of the book is the illustration of the application of statistical techniques using real-life chemical data chosen for their interest as well as what they can illustrate with respect to the associated data analysis and interpretational concepts Illustrations are explained from both exploratory data analysis and inferential data analysis aspects through the provision of detailed solutions This enables the reader to develop a better understanding of how to analyse data and of the role statistics can play within both the design and interpretational aspect of chemical experimentation I concur with the trend of including more exploratory V vi Preface data analysis in statistics teaching to enable data to be explored visually and numerically for inherent trends or groupings This aspect of data analysis has been incorporated in all the illustrations Use of statistical software enables such data presentations to be produced readily allowing more attention to be paid to making sense of the collected data I have tried to describe the statistical tools presented in a practical way to help the reader understand the use of the techniques in context I have de-emphasised the mathematical and calculational aspects of the techniques described as I would rather provide the reader with practical illustrations of data handling to which they can more easily relate and to show these illustrations based on using software (Excel and Minitab) to provide the presentational components My intention, therefore, is to provide the reader with statistical skills and techniques which they can apply within practical data handling using real-life illustrations as the foundation of my approach Each chapter also contains simple, practical, and applicable exercises for the reader to attempt to help them understand how to present and analyse data using the principles and techniques described Summary solutions are presented to these exercises at the end of the text I have not attempted to cover all possible areas of statistical usage in chemical experimentation, only those areas which enable a broad initial illustration of data analysis and inference using software to be presented Many of the techniques that will be touched on, such as Experimental Design and Multivariate Analysis (MVA), have wide ranging application to chemical problem solving, so much so that both topics contain enough material to become texts in their own right It has therefore only been possible to provide an overview of the many statistical techniques that should be an integral and vital part of the experimental process in the chemical sciences if chemical experimental data are to be translated into understandable chemical knowledge Contents xiv Glossary Chapter Introduction Introduction Why Use Statistics? Planning and Design of Experiments Data Analysis Consulting a Statistician for Assistance Introduction to the Software 6.1 Excel 6.2 Minitab Chapter 2 Simple Chemical Experiments: Parametric Inferential Data Analysis Introduction Summarising Chemical Data 2.1 Graphical Presentations 2.2 Numerical Summaries The Normal Distribution Within Data Analysis Outliers in Chemical Data Basic Concepts of Inferential Data Analysis Inference Methods for One Sample Experiments 6.1 Hypothesis Test for Mean Response 6.2 Confidence Interval for the Mean Response 6.3 Hypothesis Test for Variability in a Measured Response 6.4 Confidence Interval for Response Variability Inference Methods for Two Sample Experiments 7.1 Hypothesis Test for Difference in Mean Responses 7.2 Confidence Interval for Difference in Mean Responses 7.3 Hypothesis Test for Variability vii 10 12 16 19 21 21 26 33 34 37 42 42 47 48 51 51 53 57 59 Contents Vlll 7.4 Confidence Interval for the Ratio of Two Variances Inference Methods for Paired Sample Experiments 8.1 Hypothesis Test for Mean Difference in Responses 8.2 Confidence Interval for the Mean Difference in Responses 8.3 Hypothesis Test for Variability Sample Size Estimation in Design Planning 9.1 Sample Size Estimation for Two Sample Experimentation 9.2 Sample Size Estimation for Paired Sample Experimentation 10 Quality Assurance and Quality Control Chapter 3 One Factor Experimental Designs for Chemical Experimentation Introduction Completely Randomised Design (CRD) 2.1 Response Data 2.2 Model for the Measured Response 2.3 Assumptions 2.4 Exploratory Data Analysis (EDA) 2.5 ANOVA Principle and Test Statistic Follow-up Procedures for One Factor Designs 3.1 Standard Error Plot 3.2 Multiple Comparisons 3.3 Linear Contrasts 3.4 Orthogonal Polynomials 3.5 ModelFit 3.6 Checks of Model Assumptions (Diagnostic Checking) Unbalanced CRD Use of CRD in Collaborative Trials Randomised Block Design (RBD) 6.1 Response Data 6.2 Model for the Measured Response 6.3 Exploratory Data Analysis (EDA) 6.4 ANOVA Principle, Test Statistics, and Follow-up Analysis Additional Aspects of One Factor Designs 7.1 Missing Observations in an RBD Experiment 7.2 Efficiency of an RBD Experiment 62 63 63 69 70 73 73 76 78 80 83 84 84 85 87 90 93 94 95 99 101 101 102 107 108 110 111 111 111 112 121 121 122 356 Answers to Exercises Exercise 4.2 Based on the planned experiment, the following information is available: a = 3, b = 3, = 105.2, o2 = 464,f, = 4, f2= 9(n- I), n' = n, and k = a b = For the statistical test of interaction, 5% is the proposed significance level A starting point of n = provides a power estimate of 58% (I;= M , = 1.63) which is too low to consider Changing n to increases power to 90% cf2 = 18, = 1.99) while n = provides a power estimate of at least 96% (f2= 27 28, = 2.30) These calculations suggest at least three replications would be advised to satisfy the power constraint of at least 85% Exercise 4.3 Based on the response of percent recovery of Amaranth, the response model is p + E-123 + E-124 + E-123 x E-124 + E-120 + E-123 x E-120 + E-124 x E-120 + E-123 x E-124 x E-120 + error Recovery differs with level of E-123 tested as does consistency of results (24 mg 1-' lowest) No obvious difference is apparent for E-124 in respect of mean and variability in recovery The two levels of E-120 tested show some difference in recovery with 12 mg I-' being marginally higher Statistically, the three factor interaction is highly significant ( F = 59.43, p = 0.000 < 0.001) Follow-up through the interaction plot of the E-123 and E-124 interaction at each level of E-120 highlights different trends in the recovery data, especially at 12 mg 1-' of E-123 Diagnostic checking highlights unequal variability for all factors and non-normality for the percent recovery response as might be expected considering nature of the measured response Exercise 5.1 Based on testing linearity, it is thought the model solubility = a + fl pressure + E may fit the presented data The data, when plotted, are indicative of a positive linear trend but the points show a hint of a curved effect Fitting a linear model produces the equation solubility = - 175.104 + 2.791 pressure which is highly significant ( t = 24.94, F = 622.1, p < 0.001, R2 = 98.7%) The predicted solubility is not good with most predictions differing markedly from the measured results The predictions pattern appears to be over-, under-, and finally over-prediction suggesting presence of a trend in the results which the linear model is not explaining The pressure residual plot hints at a quadraticlcubic trend, while the normality plot looks reasonably linear The unexplained trend confirms the curved effect shown in the initial data plot Answers to Exercises 357 Exercise 5.2 The data conform almost exactly to a perfect straight line The fitted linear equation is intensity = 13.537 + 22.383 calcium which is statistically acceptable ( t = 106.3, F = 11306.8,p < 0.001, R2 = 99.9%) The average intensity measurement from the two readings is 75.9225 mV resulting in an estimated calcium value of 2.787 ppm The 95% confidence interval (5.16) is calculated to be (2.602, 2.972) ppm indicative of acceptable accuracy in the calcium concentration prediction Exercise 5.3 The plot shows an obvious non-parallel effect with the filter photometer readings rising very much more rapidly compared with the spectrophotometer readings The lines appear to cross near the origin Both fitted equations are highly significant on a statistical basis based on the following information: spectrophotometer-absorbance = 0.0 125 + 0.0022 concentration, t = 24.2, F = 584.4, p = 0.0002, R2 = 99.5%; filter photometer-absorbance = -0.0056 + 0.0071 concentration, t = 67.9, F = 4607.9, p < 0.0001, R2 = 99.9% The two equations differ in both slope and intercept Use of Test produces a test statistic (5.17) of 1277.6 (p < 0.01) indicative that the two data sets must be treated separately Test produces a test statistic (5.20) of 1292.1 (p < 0.01) providing evidence of the non-parallel nature of the two lines The point of intersection is estimated to be (3.665 pg m1-l’ 0.0204) Exercise 5.4 The MLR model fitted to the fluorescence data is fluorescence = p + p1 SVRS + p pH + p3 Heating time + p4 Cooling time + ps Delay time + E Statistically, the MLR model of fluorescence =f(SVRS, pH, HT, CT, DT) is highly significant (F = 19.42, p = 0.000 < 0.001, R2adj = 85.2%, s = 32.51) with no evidence of collinearity between the regressor variables (no correlations exceed 0.6 numerically) Both best subsets and stepwise indicate two possible ‘best’ models: the original five variable one (C, = 6) and a four variable model fluorescence =f(SVRS, pH, CT, DT) dropping HT The latter provides RZ,, = 85.5%, Cp = 4.7, and s = 32.16 corresponding to a small increase in R2adj and small decrease in s Comparing the predictive ability of each model shows that the full model is possibly better though predictions are reasonable on only a few occasions Diagnostic checking reveals no distinct trends Perhaps inclusion of interaction or power terms involving the factors could improve validity of model fit 358 Answers to Exercises Exercise 6.I The data are to be tested for method difference, a general directional difference, so the alternative hypothesis will be difference in methods (two-sided hypothesis) There is a strong overlap between the two sets of results though the Ibuprofen figures are more variable These interpretations are reflected in the similarity of medians (188.63 for Internal and 189.15 for Ibuprofen) and in the differences in the RSDs (1.7% for Internal and 3.4% for Ibuprofen) though the latter are low, indicative of consistent measurements Testing for difference in medians using the Mann-Whitney test statistic (6.2) results in acceptance of the null hypothesis indicating that there appears insufficient evidence in favour of a difference (S = 64, m = 8, T = 28, p > 0.05) Exercise 6.2 Since difference in accuracy, determined as D = Butylphenol -AFNor, is the basis of the assessment, the associated alternative hypothesis will refer to method difference (two-sided) The data plot indicates very strong similarity in the measurements of the two procedures in respect of both accuracy and precision The differences (Butylphenol - AFNor) are roughly evenly split with comparable numbers of negative and positive differences occurring with a median difference of 0.3 reported The Wilcoxon signed-ranks test statistic (6.4) is equal to 35 with n = 10 as there are no non-zero differences Comparing this with the appropriate 5% critical values leads to acceptance of the null hypothesis and the conclusion that there appears insufficient evidence of a difference in procedure accuracy (p > 0.05) Exercise 6.3 Exploratory analysis highlights that ‘pH > 8’ appears to have most effect on recovery of propazine, with the measurements collected lower, though variability appears similar for each pH level tested The summaries show different mean recoveries (89.45, 87.58, 82.63) and minor differences in RSDs (3%, 2.5%, 2.7%), the latter not sufficient for precision differences to be indicated The test statistic (6.7) is 10.85 which is significant at the 5% level Application of Dunn’s procedure using a 10% experimentwise error rate ( a = 0.1, 20.0167 = 2.1272) results in the difference between ‘pH > 8’ adjustment and the others being statistically significant Only ‘no adjustment’ and ‘pH = 5’ show no evidence of statistical difference Answers to Exercises 359 Exercise 6.4 The results obtained show no real difference between analysts with all providing similar numerical measurements though analyst C is perhaps the most consistent The median potash measurements are identical for analysts B and C and almost the same for A (15.2, 15.25, 15.25) The RSDs differ more (1.8%, 2.6%, 1.1%) indicative of different precision in the analyst’s measurements The test statistic (6.11) is 2.58 which is less than the corresponding 5% critical of x20.0s,2 = 5.99 indicating no evidence to suggest a statistical difference between the analysts Exercise 6.5 The data presented have mean 48.87%, median 50%, and standard deviation 7.94% The marginal difference between mean and median suggests normality could be in doubt The normal plot provides a waveshape which has a positive trend but which is marginally different from linearity In conclusion, it would appear there is some doubt about normality for the impurity data as might be expected considering the nature of the measured response Exercise 7.1 The normal plot of effect estimates does not conform to the hoped for trend with only the E-123 effect looking to be unimportant All other effects look important and worthy of investigation Assessing the main effects plot shows that increasing the amount of both E-124 and E-120 causes recovery of Ponceau 4R to decrease markedly Both the interactions E-123 x E-124 and E-123 x E-120 show a crossover effect indicative of differing influence on recovery of factor-level combinations Best recovery results appear to occur when all factors are set at 12 mg 1- (low level) Exercise 7.2 From the normal plot of effect estimates, effects A, A x D, D, E, and D x E stand out as very different from the unimportant effects The dotplot of effect estimates specifies the same conclusion The main effects plot highlights the positive effect of A and D and the negative effect of E on enzyme activity when increasing the factor levels For the A x D interaction, when changing levels of D, the low level of A has no effect, while the high level of A causes enzyme activity to rise A similar comparison for the D x E interaction, based on changing levels of D, 360 Answers to Exercises shows only the low level of E causing a substantive increase in enzyme activity All elements of a proposed model enzyme activity = A + A x D + D + D x E + E + error are highly significant (p < 0.01) providing evidence that the effects arrived at in the analysis are statistically important in their effect on level of enzyme activity measured Diagnostic checking revealed column length differences for A, C, D, and E suggesting important influence of these factors on variability of enzyme activity Fits and normal plots hint at two outliers, but suggest acceptable fit and response normality Exercise I The data set comprises n = 20 samples measured over p = variables PCl accounts for 32.1% of variation, PC2 22’0, PC3 17.5%, P c , 11.2%, PC, 9.2%, PCg 6.6%, and PC7 1.4% Cumulatively, at least the first four PCs would be required to explain the data adequately (82.7% contribution) For PCl, CO, NO, HC, and NO2 dominate with similar positive weights All represent pollutants generated by human activity so PCl may be a measure of human generated air pollution PC2 is dominated by SolRad, Wind, and though Wind is of opposite sign to the other two variables, suggesting that the PC may be a weather/ atmosphere component explaining how pollutants are broken down and dispersed The PC scores plot for PC, against PCI hints at some clustering of samples and 3, and 16, and 12, and and 14 though these are few in number Exercise 8.2 All samples, bar one (system B sample), appear correctly classified giving a proportion correct of 95.2% The distance measures show definite difference between systems A and B (D2= 29.0983), and A and C (D2 = 13.7607) but not B and C (D2 = 3.6344) Statistical validity of the SDA routine developed is acceptable for discrimination between A and B (F = 13.58, p < 0.01) and between A and C (F = 6.42, p < 0.05) However, validity of discrimination between B and C cannot be accepted (F = 1.70, p > 0.1) Checking mis-classifications shows that observation 12 (sample from system B) is the mis-classified observation though the difference in distance measures is small (3.335 for B and 3.083 for C) and probabilities similar (0.408 for B and 0.531 for C) One other observation (sample from system C) is a dubious classification as B and C distance measures (5.995 and 5.808) and probabilities (0.476 and 0.522) not differ substantially Predictions for the A and B specimens agree with original groupings with distance measures and Answers to Exercises 36 probabilities very different The prediction for the C specimen does predict system C but the distance measures (15.03 for B and 14.82 for C) and probabilities (0.459 for B and 0.509 for C) not differ sufficiently to be sure that the prediction is valid Subject Index Accuracy, 26,32 Adjusted coefficient of determination (R2,dj), 180, 213,217 Alternative hypothesis: see Hypotheses Analysis of covariance (ANCOVA), 167 Analysis of variance (ANOVA), 80-8 1, 82-83,90-92,112,138-140, 157- 158,178- 179,243,280-28 1, 326 mean squares (MS) in, 90, 112, 139-140,158, 178,212,281 sums of squares (SS) in, 90, 112, 139-140,157, 178,212,280,326 Analysis of variance table for CRD, 91,108 for linear regression, 178 for MLR, 213 for RBD, 112 for three factor factorial design, 158 for two factor factorial design, 139 Ansari-Bradley test, 234 Assumptions in factorial designs, 138, 157 in one factor designs, 85-87, 111,242, 250 in one sample inference, 43 in paired sample inference, 64, 236 in regression modelling, 173, 21 in SDA, 315,316 in two sample inference, 52,230 Bartlett’s V statistic, 325-326 Best subsets multiple regression: see MLR Blocking, 1-82, 110- 11 British Standards Institution (BSI), 77,79 C, statistic, 21 Calibration: see Linear calibration Chemical experimentation modelling, 2-3 monitoring, optimisation, Chemical significance, 42 Chemometrics, 3,293 Chi-square (x2) test statistic for SDA, 325,326 for variance, 48-49 Cluster analysis, 328 Coefficient of determination (R2),179, 180 see also Adjusted coefficient of determination (R2,dj) Coefficient of variation (CoV): see RSD Collaborative trials (CT), 108-1 10 Comparison of analytical procedures: see Comparison of linear equations Comparison of linear equations, 195-206 single versus separate, 200-202 test of parallelism, 202-203 Completely randomised design (CRD), 83-1 10 design structure, 84, 107 diagnostic checking, 102- 106 follow-up analysis, 93- 102 main analysis, 85-93 non-parametric alternative, 241 -249 power estimation, 123-125 response model, 84-85 sample size estimation, 125- 126 unbalanced, 107- 108 use in collaborative trials, 108- 110 Confidence intervals, 41-42 in collaborative trials, 109 in linear calibration, 191 362 Index in linear contrasts, 100,249 in linear regression for slope, 179 in one sample inference for mean, 47 for variance, 51 in paired sample inference for mean difference, 69 for median difference, 240 in two sample inference for difference of means, 57 for difference of medians, 234 for ratio of variances, 62 Consulting a statistician, 8-10 Corrected sum of squares, 27, 174 Correlation in multi-collinearity assessment within MLR, 13 in paired sample inference, 70 Correspondence analysis, 328 Data matrix in MVA, 296 Data plots boxplot, 22 dotplot, 21-22 interaction, 134, 147 main effect, 95,275 normal probability, 258-259 profile: see interaction residual, 102- 104 scatter, 171 standard error, 94 Data reduction, 295 see also Correspondence analysis, Factor analysis, PCA Data summaries mean, 26 median, 26 median absolute difference, 37 quartiles lower, 22 upper, 23 range, 27 relative standard deviation (RSD), 27 skewness, 23 standard deviation, 27 variance, 28 Data transformations, 127- 129 Decision rule in inferential data analysis, 40-41 363 Degrees of freedom, 27 Descriptive statistics, 20 Design of experiments, 5-7 Detection limits, 195 Diagnostic checking in experimental designs, 102- 104, 150, 283-284 in regression, 182- 183,219-220 Distance measures in MVA, 298-299 Euclidean, 298-299 Mahalanobis, 299 Distribution-free methods: see Nonparametric methods Dummy variables: see Multiple non-linear regression Dunn’s multiple comparison procedure, 246-247 Effect sparsity, 268 Eigenvalue equation, 297 Eigenvalues and eigenvectors in MVA, 297-298 in PCA, 300-301 in SDA, 314 Errors in inferential data analysis: see Type I error, Type I1 error Estimation: see Confidence intervals Excel software, 11- 16 dialog windows, 13, 28, 53, 60,65, 175 Experimentwise error rate, 96,246 Exploratory data analysis (EDA), 43, 52, 64,87,111, 138,230,236,242,250, 27 Extreme values: see Outliers F test statistic in factorial designs, 140, 158, 164, 281 in one factor designs, 90-9 1, 113 in orthogonal polynomials, 101 in regression modelling, 179, 20 1,202, 21 3,224 in SDA, 316 in two sample inference, 59 Factor analysis, 312-313 Factorial designs, 132- 164 design structure, 136-1 37 diagnostic checking, 150 follow-up analysis, 146- 149, 159 main analysis, 138- 146, 157- 159 method validation, 156 364 no replication, 155 non-parametric alternative, 256-257 overview of data analysis, 152, 159 pooling of factors, 164 power estimation, 152- 153 pseudo-F test, 164 response model, 137-1 38, 157 sample size estimation, 152- 153 two-level: see Two-level factorial experiments unbalanced, 155 Factors: see Treatments False negative: see Type I1 error False positive: see Type I error Fisher’s linear discriminant function, 14 Fractional factorial designs, 288-290 Friedman’s test, 250-251 Heteroscedasticity, 103, 127 Hierarchical designs: see Nested designs Homoscedasticity, 127 Hypotheses, 37-39 alternative, 37-38 in experimental designs, 90,97, 100, 138, 157,242 in multiple comparisons, 97, 247,254 in one sample inference, 43,49 in paired sample inference, 64, 236 in regression modelling, 179, 187 in SDA, 316 in two sample inference, 52, 59, 230 null, 37-38 one-sided, 38 two-sided, 38 Incomplete block design, 130- 131 Inferential data analysis, 20, 37-42 errors, 39-40 estimation, 41-42 hypotheses, 37-38 p value, 40-41 power, 40 practical significance, 42 one-tailed test, 38 test statistic, 39 two-tailed test, 38 Inner fences in boxplots, 22 Interaction in experimental designs, 133- 134, 146- 148 Index in regression modelling, 223, 224 Intercept in linear regression in linear calibration, 191 test of, 187-188 zero in linear modelling, 188- 189 International Standards Organisation (ISO), 77,79 Intrinsically linear models: see Non-linear modelling Intrinsically non-linear models: see Nonlinear modelling Inverse prediction: see Linear calibration Kruskal-Wallis test, 241-243 Laboratory of the Government Chemist (LGC), 4, 19,78 Latin square design, 129- 130 Least squares estimation ordinary (OLS), 173-1 74,21 I weighted (WLS), 225 Linear calibration, 189- 195 confidence interval for estimated X value, 191 test of line through origin, 191 test of sensitivity, 191 Linear contrast, 99-101 in two-level designs, 265-267 non-parametric alternative, 249 Linear regression, 171- 189 diagnostic checking, 182- 183 model, 172- 173 no intercept, 188-189 non-parametric alternative, 257-258 parameter estimation, 173- 174 practical validity, 182 scatter plot, 172 statistical validity, 179- 180 test of intercept, 187- 188 test of slope against target, 186- 187 Main effects, 133, 265 Mann-Whitney test, 230-23 Mean: see Data summaries Mean squares (MS) in factorial designs, 139- 140, 158 in one factor designs, 90, 12 in regression, 178, 212 Measures of location, 26- 27 Measures of variability, 27 28 365 Index Median: see Data summaries Median absolute difference (MAD): see Data summaries Minitab statistical software, 16- 18 dialog windows, 86,91,94, 147, 212, 269,301,315 Mixture experiments, 29 1-292 Model building in regression, 206-209, 223 Multi-collinearity: see Correlation Multiple comparisons, 95-99,245-249, 254-256 non-parametric procedures Dunn’s procedure, 246-247 procedure associated with Friedman’s test, 254 Student-Newman-Keuls (SNK) procedure, 96-97 Multiple linear regression (MLR), 209-222 best subsets, 216-218 comparison of models, 224-225 diagnostic checking, 219-220 model, 21 multi-collinearity, 213 parameter estimation, 21 practical validity, 219-220 statistical validity, 213 stepwise methods, 216-21 Multiple non-linear regression, 223- 224 Multivariate analysis (MVA) methods, 293-299 National Measurement and Accreditation Service (NAMAS), 4-5, 79 Nested designs, 164- 166 Non-central F distribution, 122- 123 Non-linear regression, 207-208 intrinsically linear models, 207-208 intrinsically non-linear models, 208 Non-parametric methods, 227-262 in linear regression, 257-258 in one factor designs, 241-255 in paired sample inference, 235-240 in two factor factorial designs, 256-257 in two sample inference, 229-234 Normal distribution, 33-34 Normal equations for linear regression, 173 for MLR, 21 Normal probability plot for testing data normality, 258-261 of residuals, 103-104 Normality tests Anderson-Darling test, 262 Lilliefors test, 26 Ryan-Joiner test, 261-262 Null hypothesis: see Hypotheses One-factor-at-a-time experimentation (OFAT), 2,132- 133,135- 136 One-tailed test: see Inferential data analysis One factor designs: see CRD, RBD One sample inferential data analysis, 42-51 confidence intervals for mean, 47-48 for variability, 51 tests x2 test for variability, 48-50 t test for mean, 42-47 Orthogonal polynomials, 101 Outer fences in boxplots, 35 Outliers, 34-37 p value, 40-41 Paired sample inferential data analysis, 63-72,235-240 confidence intervals for mean difference, 69 for median difference, 240 tests paired comparison t test, 63-69 t test for variability, 70-72 Wilcoxon signed-ranks test for median difference, 235-237 Parallel lines in comparison of linear equations, 196 in interaction, 134 Parameter estimation in regression for linear model, 173- 174 for MLR model, 21 Partial least squares (PLS), 312 Polynomial regression, 208-209 Pooled standard deviation, 52 Pooled t test, 52 Power, 40 Power analysis, 73-78, 122- 127, 152-1 55 see also Sample size estimation 366 Precision, 27,28, 32 Prediction in linear calibration, 191 in regression, 182, 219 in SDA, 316 in two-level factorial designs, 282 Principal component analysis (PCA), 299-3 10 number of PCs, 302 objective of, 300-302 PC scores, 307 PC weights, 305 Principal components regression (PCR), 311-312 Quadratic regression model: see Polynomial regression Qualitative data, 20 Quality assurance, 78-79 Quality control, 78-79 Quantitative data, 20 Randomisation, Randomised block design(RBD), 111-122 design structure, 10- 11 diagnostic checking, 113 efficiency, 122 follow-up analysis, 113 main analysis, 11 1- 13 missing observations in, 121- 122 non-parametric alternative, 250-255 power estimation, 123- 125 response model, 11 sample size estimation, 125- 126 Ranking of data, 227-229 Range: see Data summaries Regression modelling, 168-226 Relative standard deviation (RSD): see Data summaries Repeatability, 32 Repeated measures design, 166- 167 Replication, in two-level factorial designs, 287-288 Reproducibility, 32-33 Response model, 82 in factorial designs, 137- 138, 157, 280 in one factor designs, 84-85, 11 in regression, 172- 173,207,208,2 11, 224 Index Response surface methods (RSM), 290-29 Residual analysis: see Diagnostic checking Residual mean square, 173,225 Sample mean: see Data summaries Sample size estimation in factorial designs, 152- 153 in one factor designs, 125-126 in paired sample experiments, 76-77 in two sample inference, 73-75 Screening experiments: see Fractional factorial designs Separate-variance t test, 52-53 Significancelevel, 39 Skewness, 22-23 Smoothing, 226 Sorting and grouping, 295-297 see also Cluster analysis, SDA Spline functions: see Smoothing Standard deviation: see Data summaries Standardisation of data, 297 Statistical discriminant analysis (SDA), 313-328 a priuri probabilities, 327-328 analysis elements, 314-3 17 discriminant scores, 314 misclassifications, 16 objective of, 313-314 predictive ability, 16 statistical validity, 16, 325-326 stepwise methods, 327 traininghest set, 327 Stepwise multiple regression, 216-21 Student-Newman-Keuls (SNK) multiple comparison, 96-97 Studentised range statistic, 97 Student’s t distribution, 43-44 Sum of squares (SS) in factorial designs, 139- 140, 157, 280-28 in one factor designs, 90, 12 in regression, 178,212 Symmetric data, 22,34 t test statistic in one sample inference, 43 in paired samples inference, 64,70 in regression, 179, 187, 205 in two sample inference, 52 367 Index Test statistic, 39 Three factor interaction, 158, 159 Treatments, 80, 135 Two-level factorial experiments, 263-288 centre points, 288 contrasts, 265-267 data analysis, 275-276 effect estimate plots active contrast, 274 dotplot, 274 half-normal, 274 normal 272 pareto chart, 274 effect estimation, 267-268 modelling in, 280 prediction in, 282 statistical assessment in, 280-284 Two sample inferential data analysis, 1-62,229-234 confidence intervals for difference in means, 57-58 for difference in medians, 234 for ratio of variances, 62 tests Ansari-Bradley test for variability, 234 F test for variability, 59-62 Mann-Whitney test for difference in medians, 230-23 t test for difference in means, 52-57 Two-tailed test: see Inferential data analysis Type I error, 39-40 Type I1 error, 39-40 Unbalanced data, 107-108,121-122,155, 247 Uncontrolled variation, 82 United Kingdom Accreditation Service (UKAS), 5,78-79 Valid Analytical Measurement (VAM), 4, 78 Variability, 27 Variance: see Data summaries Variance-ratio test, 59-60 Weighted least squares (WLS), 225 Welch’s test: see Separate-variance t test Weighted principal component analysis (WPCA), 310 Wilcoxon rank sum test: see MannWhitney test Wilcoxon signed-ranks test, 235-237 Wilks’ A statistic, 326 z score, 35 [...]... data to help gain an initial insight into the structure of the data Factor analysis (FA) An MVA data reduction technique for detection of data structures and patterns in multivariate data Heteroscedastic Data exhibiting non-constant variability as the mean changes Homoscedastic Data exhibiting constant variability as the mean changes Inferential data analysis Inference mechanisms for testing the statistical. .. the name given to the cross-disciplinary approach of using mathematical and statistical methods to help extract relevant information from chemical data The increased power and availability of computers and software has enabled statistical methods to become more readily available for the treatment of chemical data On this basis, all analysis concepts will be geared to using software (Excel and Minitab)... these are to be displayed and analysed (statistical analysis methods) Design and statistical analysis must be considered as one entity and not separate parts to be put together as necessary A well planned experiment will produce useful chemical data which will be easy to analyse by the statistical methods chosen A badly designed and planned experiment will not be easy to analyse even if statistical methods. .. measure of the absolute precision of replicate experimental data Statistical discriminant analysis (SDA) An MVA sorting and grouping procedure for deriving a mechanism for discriminating known groups of samples based on measurements across many common characteristics Systematic error Causes chemical measurements to be in error affecting data accuracy Test statistic A mathematical formula numerically... cross-disciplinary approach of using mathematical and statistical methods to extract information from chemical data Cluster analysis An MVA sorting and grouping procedure for detecting well-separated clusters of objects based on measurements of many response variables Confidence interval An interval or range of values which contains the experimental effect being estimated xiv Glossary xv Correspondence analysis An... are relevant to all aspects of experimentation ranging from planning to interpretation The latter can be subjective (exploratory data analysis, EDA) as well as objective (inferential data analysis, estimation) but the basic rule must be to understand the data as fully as possible by presenting and analysing them in a form whereby the information sought can be readily found Examples where statistical. .. chemical implications of the results 6 INTRODUCTION TO THE SOFTWARE Spreadsheets, such as Excel,5 and statistical software, such as Minitab,6 are important tools in data handling They provide access to an extensive provision of commonly used graphical and statistical analysis routines which are the backbone of statistical data analysis They are simple to use and, with their coverage of routines, enable a. .. statistics in scientific experimentation3 together with a greater level of usage Use of statistical techniques are advocated by professional bodies such as The Royal Society of Chemistry (RSC) and the Association of Official Analytical Chemists (AOAC) for the handling and assessment of analytical data to ensure their quality and reliability Statistical procedures appropriate to this type of approach. .. Minitab is best as it is simple to use and compatible in most of its operation with the operational features of Excel The data presentation principles I will instil can be easily carried forward to other software packages In the statistical data analysis illustrations, I will present and explain briefly the dialog window associated with the analysis routine for the software being used to generate analysis. .. therefore likely to make the conclusions more applicable 5 CONSULTING A STATISTICIAN FOR ASSISTANCE Many experimenters believe that a statistician’s role is only to help with the analysis of data once an experiment has been conducted and data collected This is fundamentally wrong A statistician can provide assistance with all aspects of experimentation from planning through to data analysis so that the ... Statistical Analysis Methods for Chemists A Software- based Approach Statistical Analysis Methods for Chemists A Software- based Approach William P Gardiner Department of Mathematics, Glasgow... cross-disciplinary approach of using mathematical and statistical methods to extract information from chemical data Cluster analysis An MVA sorting and grouping procedure for detecting well-separated clusters... and software has enabled statistical methods to become more readily available for the treatment of chemical data On this basis, all analysis concepts will be geared to using software (Excel and